Posts Tagged ‘Existence’

Proof Of Existence Of Square Root Of Two For Ten Year Old (Take That, Euclid!)

September 6, 2020

Yesterday my ten year old daughter cried like I have not seen her cry since her grandmother died. Tears were running down her cheeks:”Cela n’a aucun sens!” It makes no sense! It was a telling choice of words: the Greek mathematicians made the exact same observation, they called such numbers “irrational”. I reassured her: she was brutally confronted by what drove ancient Greek mathematicians crazy: the square root of two.

How to define square roots algebraically is simple: x is the square root of a if and only if: xx = a… No problem if a is one, four, nine… But put a = 2, and a mystery arises: one can write down numbers ever closer to x = square root of two, but then what? Their squares are never quite two… Worse: ancient Greek mathematicians knew (positive) integers, and their ratios, say m/n, where both m and n are integers. They could demonstrate, in a few lines, that square root of two was not a fraction. 

Mathematics is the way of thinking that is most powerful in some situations. For example, as I wrote: xx = 2, I used no less than three mathematical notions that the ancient Greeks mathematicians did not have: the notion of equation, the equal sign, and the notion of unknown x.  

When the Greeks were confronted to this, they realized that some things existed that should not exist, according to their (mathematical) system of thought. Obviously, their mathematics came short. They discovered “irrational” numbers, the hard way. Further reasonings were halted by the rise of fascism in the Hellenistic and then Roman world. Do we have similar situations nowadays? Obviously yes. Nonlocality in Quantum Physics is an obvious example.

There is a mysterious relationship between numbers and geometry. Draw a line: that’s the x axis. Some distance mark one (1); it is the unit of distance. It does more: it represents the number one. Draw the perpendicular axis, traditionally called the y axis. There too, mark a one. Use the proverbial compass to make the units on the x and y axis the same length. Now one is facing a two dimensional plane. It naturally defines a unit of area, the square of sides equal to one. Cut that square in two to get the area of surface ½, etc… A bit of playing around shows that any square of side s has area ss…

Consider the rectangular triangle from the origin with sides one. The longest side can be used as the base of a square. A quick look shows that this square has area two (2). Thus its side is square root of two. 

This demonstrates that the square root of two exists. 

Ancient Greek mathematicians went that far. And that drove them nuts, as, for them, any “number” had to be of the form m/n, with m and n integers… And the square root of two was not such. 

The reasoning above is mine: it was designed to be understood by a very curious ten year old who is not fanatical about mathematics. So the Pythagorean theorem is demonstrated, in a particular case (the general proof is a somewhat confusing generalization; similar ad hoc geometrical proofs work for other numbers, for example square root of 5). 

Patrice Ayme



Why the Greeks knew square root of two was no fraction (= “rational” number):

If (m/n)^2 = 2, with xx = x^2 and m and n having no common factors… Then:

mm = 2 nn… Thus m is even (if m is odd, mm is always odd). So m = 2u. This forces n to be odd, because by hypothesis, m and n don’t have common factors, so can’t both be divided by 2…

Hence, n = 2v + 1, for some integer v… Plugging back in the initial equation mm = 2 nn we get: 2u2u = 2 (2v+ 1) (2v + 1). Dividing by 2, we get:

2uu = (2v + 1) (2v +1) = 4vv + 4v + 1…

Now that latter equation is impossible: the left side is even, and the right side is odd… Thus the initial hypothesis, (m/n) (m/n) = 2 is impossible…

Of God, Mice, And Men Who Believe They Created The Universe

February 8, 2018

When theists say that the universe exists because of God, they are saying that the universe exists, because of some agent they know: that make those theists vastly superior to us, simple miscreants, who do not happen to be acquainted with what, or who, created all and everything. Surely, those superior beings should lead us? So what sounds metaphysical, by asserting a “God” boils down to claiming a higher place in an all too human hierarchy.

Universe” means literally, “turned into one”, whereas “multiverse” would be: “turned into many”. So the set of all multiverses is the universe. (So the alleged existence of “multiverse” is akin to Bertrand Russell’s famous paradox of the set whose elements are not elements of itself; Russell’s paradox brought down mathematical logic as it had been known prior; present day physicists have been repeating that mistake, from lack of basic culture in the matter of mathematical logic!)

If we were to claim, and, or, even worse, have the feeling, that we know why the universe exists, we would be claiming, or have the impression, that we were God. This is not the business of physics, only the business of those who want us to be guided by absolutism.

Alexander the Great, seeing his blood flow, asked himself that question: am I a God? His Greek and Macedonian companions laughed him off. Later, on the advice of his mom, Olympia, Alexander ordered the old, most senior generalissimo Antipater, a companion of Alexander’s father, from Greece to Babylon. Antipater refused to obey. Antipater’s youngest son was Alexander’s page. Alexander found himself ceasing to be, before he could even organize his affairs.

We are both everything and nothing relative to the universe. The key to wisdom, is to keep a balance.

Man, playing God, touches man, playing Adam. All very touching, self-obsessing, self-gratifying, self-glorifying mental, self-stimulation, and self-mutilation.

The universe is, what it is. Science can describe it, not explain how it came to be. That is the proper mood that wisdom should embrace. Embracing the humility of reality, so we can unleash the power of truth.

Let theologians, dinosaurian conservatives, the Politically Correct and the Perfect Cretins, among others, try to learn this: We have to embrace the way things are, before we can hope to change what needs to be changed. And there is plenty of the latter. So stop claiming some human beings know why there is all there is. They don’t. They, and, or, their supporters just want everything you could possibly imagine, and then more.

Patrice Aymé

Note 1: the comment above was an answer to: “Why Is There Something, Rather Than Nothing?
Posted on February 8, 2018 by Sean Carroll
A good question!

Or is it?”

In it, Sean points out notions which I have exposed in the past, but are worth repeating, as many physicists, let alone philosophers and theologians, don’t get them. First of all Sean basically points out that the universe just is (as I said above, by definition of this neuronal activity!). And secondly Sean Carroll, a famous Cal Tech cosmologist, points out that all too many professional physicists don’t even understand that physics, as presently understood, doesn’t explain the universe! In other words, as I have said for decades, all too many physicists take themselves for God! (That is in the same meta category as Niels Bohr’s famous retort to Albert Einstein:”Stop telling God what to do!“)

“The right question to ask isn’t “Why did this happen?”, but “Could this have happened in accordance with the laws of physics?” As far as the universe and our current knowledge of the laws of physics is concerned, the answer is a resounding “Yes.” The demand for something more — a reason why the universe exists at all — is a relic piece of metaphysical baggage we would be better off to discard.

This perspective gets pushback from two different sides. On the one hand we have theists, who believe that they can answer why the universe exists, and the answer is God. As we all know, this raises the question of why God exists; but aha, say the theists, that’s different, because God necessarily exists, unlike the universe which could plausibly have not. The problem with that is that nothing exists necessarily, so the move is pretty obviously a cheat. I didn’t have a lot of room in the paper to discuss this in detail (in what after all was meant as a contribution to a volume on the philosophy of physics, not the philosophy of religion), but the basic idea is there. Whether or not you want to invoke God, you will be left with certain features of reality that have to be explained by “and that’s just the way it is.” (Theism could possibly offer a better account of the nature of reality than naturalism — that’s a different question — but it doesn’t let you wiggle out of positing some brute facts about what exists.)

The other side are those scientists who think that modern physics explains why the universe exists. It doesn’t! One purported answer — “because Nothing is unstable” — was never even supposed to explain why the universe exists; it was suggested by Frank Wilczek as a way of explaining why there is more matter than antimatter. But any such line of reasoning has to start by assuming a certain set of laws of physics in the first place. Why is there even a universe that obeys those laws? This, I argue, is not a question to which science is ever going to provide a snappy and convincing answer. The right response is “that’s just the way things are.” It’s up to us as a species to cultivate the intellectual maturity to accept that some questions don’t have the kinds of answers that are designed to make us feel satisfied.”

Note 2: Swiss citizen Tariq Ramadan, the world’s most famous  Islamist propagandist, holder of two chairs (no less!) at Oxford University, and now in a French prison, was going around the world grievously beating and raping women. Why? Because, precisely, he wanted everything, and that included beating up handicapped women. Even now, as he sits in prison, he enjoys his power: immensely powerful organizations behind him, the sort who made him an Oxford Don, are threatening many more women, who also want to file complaints against Ramadan, but are afraid to do so. The human species is naturally metaphysical. Ramadan wanted to create a universe where he and his ilk could hurt and terrorize women at will. This is not any different from telling us that Muhammad flew to Jerusalem, the capital of Israel, on a winged horse: it is outrageous, but it creates a universe, and its cause (and in this case Islamists are the cause of said universe!)

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